The Sharpe model, unlike the Markowitz model, requires less information and calculations. Sharp concluded that the returns of each individual stock are highly correlated with the overall return of the market, so there is no need to determine the covariance of each stock with each other, just determine how they interact with the market.
The Sharpe model is based on the method of linear regression analysis, which allows one to relate two random variables - independent (X) and dependent (Y) by the linear expression Y = α + β·X. In the Sharpe model, the expected return on stock market overall (market portfolio return) Rm, calculated based on the Standard and Poor’s index. The dependent variable is the profitability Ri of some securities. Let the profitability Rm take random values Rm1; Rm2…. Rmn, and the yield of the i-th security is Ri1; Ri2…. Rin. Then the linear regression model representing the relationship between the market return and the return on a specific security will have the form:
Ri = αi + βi Rm + εi,
where Ri is the yield of the i-th security at a certain point in time (for example, June 25, 2003);
αi is a parameter showing what part of the profitability of the i-th security is not associated with changes in the profitability of the securities market Rm;
βi is a coefficient showing the sensitivity of the yield of the i-th security to changes in market yield;
Rm is the return on the market portfolio in this moment time;
εi is a random error due to the fact that the actual values of Ri and Rm sometimes deviate from a linear relationship. To simplify calculations, it can be taken equal to 0.
βi - beta coefficient - a measure of investment risk, reaction (sensitivity) of the expected income on a security to a change external factors;
βi = σi , βi = ρi,m σi ,
where σi is the standard deviation of the yield of the i-th security;
σm - standard deviation of profitability for the market as a whole;
ρim is the correlation coefficient between the yield of the i-th security and the market as a whole.
Assuming that the investor is constructing a portfolio of n securities, Sharpe introduces the following preconditions:
The arithmetic mean of random errors Eε for all securities in the portfolio is equal to 0;
The variance of random errors σε² for each security is constant.
For each security, there is no correlation between the random error values observed over T years;
There is no correlation between random errors εi and market returns;
There is no correlation between the random errors of any two securities in the portfolio.
Based on these simplifications, Sharpe, for any securities in the portfolio, obtains the following expressions:
Еi = αi + βi Em ,
σi² = βi² · σm² + σεi² ,
σij = βi² βj² · σm² ,
where Ei is the expected arithmetic average return of securities i;
Em is the expected arithmetic mean return of the market portfolio;
σi² - dispersion of the i-th security;
σm² - dispersion of the market portfolio;
σεi² - random error variance;
σij (covij) is the covariance between the yields of security i and security j;
βi and βj are the sensitivity of the yield of the i-th and j-th security to changes in market yield.
Thus, to construct boundaries of efficient portfolios there are all the necessary elements: Ei; σi²; σij.
Expected portfolio return, consisting of n securities, is calculated by the formula:
Ep= ∑ Хi Еi
Portfolio variance in the Sharpe model it is represented as:
σn² = ∑ Хi² σεi² ,
σεi² = ∑ (Rit - (αi + βi Rmt)) ² / (n-2)
Self-test questions
1. What is a securities portfolio?
2. Describe the different types investment portfolios.
3. Describe an aggressive, conservative and moderately aggressive investor.
4. What is meant by active and passive investment portfolio management?
5. What is investment portfolio diversification?
6. How to determine the profitability and risk of an investment portfolio?
7. What does positive and negative covariance mean between securities returns?
8. What does the correlation coefficient characterize?
9. What is the effective Markowitz frontier?
10. How is the return on a company’s securities and β – the coefficient in the Sharpe model calculated?
Good afternoon, dear community of traders, investors and everyone interested in the securities market!
W. Sharp's model or as it is often called, the market model was first proposed by an American economist, Nobel Prize winner William Forsythe Sharp in the mid-60s of the last century.
William F. Sharp is currently Professor Emeritus at the Graduate School of Business at Stanford University.
In 1990 he received the Nobel Prize in Economics, which he received for his development of valuation theory financial assets.
The Sharpe model represents the relationship between the expected return of an asset and the expected return of the market. It is assumed that the return on an ordinary share for a certain period is related to the return over similar period with profitability market index. In this case, as the market index rises, the share price is likely to rise and vice versa.
Thus this model is assumed to be linear. And the equation of the proposed model has the following form: 
The main difference between W. Sharpe’s model and G. Markowitz’s model is as follows:
The Sharpe model examines the relationship between the returns of each security and the returns of the market as a whole, while the Markowitz model considers the relationship between the returns of securities.
It was in order to avoid the high complexity of the Markowitz model that William Sharp proposed a market (index) model. At the same time, the Sharpe model is not a new method of compiling a securities portfolio - it is a simplified Markowitz model, where the solution to the problem of choosing the optimal portfolio is carried out with less effort. The Sharpe model is usually used when considering large quantity securities that represent a significant part of the market.
It is very interesting to compare the results obtained using the Markowitz model and the Sharpe model.
For this purpose, I developed an application in Microsoft Office Excel* called "".
In my recent post, I demonstrated the result of calculating the determination of the optimal calculation for Russian market shares according to the Markowitz model with the following inputs:
- shares included in the calculation of the main index of the Moscow Exchange - the MICEX Index - 50 of the most liquid and capitalized securities on the Russian stock market were taken;
- the historical period for analysis for the instruments under consideration was selected from January 9, 2007 to October 24, 2013;
- level of expected profitability - maximum;
- the level of acceptable risk is minimal;
- diversification (maximum share of investments in a financial instrument) - 15% of existing assets;
- The minimum level of daily liquidity for shares is 6 million rubles.
You can see the results obtained for these models below:
Markowitz model: 
Sharpe model: 
As you can see, the difference in the composition of the proposed optimal securities portfolios is small. In the Sharpe model, the share of Severstal securities was 11% versus 2.8% in the Markowitz model; Bashneft shares in the Sharpe model are less than 1%, in the Markowitz model - 5.8%; in the Sharpe model, NLMK shares are -13.3%, in the Markowitz model - 15%; in the Sharpe model there are no Tatneft shares at all, in the Markowitz model - 1.5%. The remaining shares of papers are the same for the described models.
The final parameters are as follows:
Markowitz model: 
Sharpe model:
Here we observe that, with the same level of risk, the return on Sharpe’s portfolio turns out to be slightly higher than the return on the Markowitz model - 26.75% versus 24.32% per annum, respectively. At the same time, we see that the beta of the portfolio according to the Sharpe model is also higher than the beta obtained according to the Markowitz model (0.64 versus 0.59), and this, in turn, suggests that the Sharpe portfolio is slightly less defensive (protective) than Markowitz's briefcase.
W. Sharpe's market model of the optimal portfolio ultimately looks like this:
All other calculated indicators are in the presented application " Portfolio investments in the Russian stock market according to the W. Sharpe model (market model)" are the same as in the Markowitz model. 
Application " Portfolio investments in the Russian stock market according to the W. Sharpe model (market model)"also contains the same technical characteristics as the application "Portfolio investments in the Russian stock market according to the Markowitz model"
The transaction journal has a convenient, quick transition from one page to another using internal hyperlinks. Hyperlinks to graphs will allow you to quickly go to the desired summary table on the basis of which they are built. In stock detailed instructions to work with the application.
In total there are more than 65 different charts, more 75 pivot tables and everything is clearly structured.
The application is configured so that you can easily print all the sheets (there is no need to specially format them) in order to make special folders for yourself where you can file your calculations, etc. and so on. All pages are numbered.
You can also, if desired, convert it into a convenient, readable PDF format (if available special program to create PDF files).
For clarity, I posted the final data file, converted to PDF format, on a shared disk. You can follow the link and watch or download:
All formulas in the application are open so that you can look into the depths of the calculations themselves in terms of the various indicators used in the application.
If desired, the original database of the application about the price parameters of the financial instruments already included in it can be changed, expanded (both according to the list of securities under consideration and according to the horizon of their research) and, of course, periodically update the application for the current date.
In conditions of developed and stably functioning stock markets, the above-mentioned classic models Markowitz and Sharpe work quite effectively. At the same time, in modern conditions Using just one model alone is not correct. The models of W. Sharpe and G. Markowitz can be a good addition to other factors when compiling an optimal portfolio of securities.
"Portfolio investments in the Russian stock market according to the W. Sharpe model (market model)" is an excellent tool for a professional approach to investing in the securities market.
If you are interested in the application, you can purchase it either on the website.
The Sharpe ratio is an indicator of the efficiency of an investment portfolio (asset)
The Sharpe ratio shows the efficiency of the investment portfolio and is calculated using the formula
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Sharpe ratio is the definition
Sharpe ratio- This an indicator of the effectiveness of an investment portfolio (asset), which is calculated as the ratio of the average risk premium to the average deviation of the portfolio. In other words, we can say that the Sharpe ratio is the mathematical ratio of the average return to the average deviation of this return.
Sharpe ratio is a kind of indicator of the effectiveness of the system. The higher it is, the more profit the system will bring. The Sharpe ratio is rarely higher than one, and this happens mainly when determining efficiency in banking system. In this case, the system will show returns with maximum profit.
Sharpe ratio is return-risk relationship. This ratio indicates the possible degree of stability of the expected profit.
Sharpe ratio is designed to understand how much the return on an asset compensates for the risk taken by the investor. When comparing two assets with the same expected return, investing in the asset with the higher Sharpe ratio will be less risky.
Options for calculating the Sharpe ratio
There are many options for calculating the Sharpe ratio, but they are all based on the same idea:
Sharpe Ratio = (Return – Risk Free Return)/ Standard Deviation of Return
Note that the right-hand side can be expressed in either dollars or percentages, as long as both sides of the equation are expressed in the same units. A few words about certain terms that are best expressed in annual terms:
1. Profitability. This is the amount you earn from your assets.
2. Risk-free Return. This is the amount of money you can expect to earn from assets that are economic analysis are classified as “risk-free”, for an amount of capital equivalent to that with which you intend to enter the market where you operate. In all situations, with few exceptions, the appropriate rate of return here will be the rate at financial instruments US Treasury. When calculating the Sharpe ratio, the risk-free return is subtracted from the total return of the portfolio to isolate the portion of the indicator that is tied to the assumption of market risk exposure. One of the rather neat results here is that someone who takes capital and invests it in Treasuries earns exactly the risk-free interest rate, and therefore the Sharpe ratio in this case becomes zero, and for those portfolios that do not can bring even such a modest level of profitability, the Sharpe ratio will be negative. Therefore, the Sharpe ratio becomes positive only when the achieved indicators are higher minimum rate for government securities - that is, in principle, it is assumed that these indicators are associated with some risky market activity, and then we can talk about positive risk-adjusted returns.
3. Standard deviation of Return. This old friend of ours: we thought we had smashed him to smithereens, but no; here it is, right there, rising from the ashes to participate as a risk component in the calculation of risk-adjusted returns. Note to yourself that it is extremely important to express this statistical value for the appropriate period of time - ideally, as mentioned above, for one year. Due to the specifics of this calculation (when this figure varies directly with the square root of the number of partial values of observations), this requires either multiplication or division of the square root of the number of observations. For example, suppose you have daily data for a year that defines a daily standard deviation of, say, $10,000, or 1% (let the equity amount be $1 million). To find the annualized standard deviation, you need to multiply this figure by the square root of the quantity operating days per year. If you cross out weekends and holidays on your calendar, you get about 250 plus or minus a day or two, and the square root of that number is about 15.9. Therefore, if the daily standard deviation is $10,000, or 1%, then the annualized standard deviation will be approximately $159,000, or 15.9%.
In the formula for calculating the Sharpe ratio, such normalization by time intervals must be performed in order for the results to be meaningful. Note that this formula allows for adjustments to account for factors such as the fact that the data set may not be complete (for example, six months of data) and that the time periods will not necessarily be one day. However, in my explanations of these mysterious phenomena, I will rely on the opinions of my friends who are professional in the field of statistics.
By now, you've probably rushed to calculate your Sharpe ratio, and you're wondering whether you should be ashamed or proud of your results. As a simple rule of thumb, I think you should almost always aim for the Sharpe ratio calculated using the above method to be greater than or equal to one. For example, if we assume that the risk-free interest rate is 5%, and the standard deviation of annualized returns is 15%, then to achieve this threshold, such a portfolio would need to have a return of at least 20%:
(20% return - 5% risk-free interest rate) / 15% return standard deviation = 1.0
Of course, if Sharpe ratio less than this basic value, then you can still achieve fairly high financial goals over long periods of time; however, the attractiveness of such returns from a risk-adjusted perspective is naturally reduced. In such cases, the capital provider (whether you yourself or some other economic entity), quite rightly, will come to the conclusion that his money can be used for more interesting purposes. At the other extreme, I have seen some portfolios reach Sharpe ratios of 5.0, 10.0, or even higher over long periods of time. Such - quite rare - exceptions may indicate either an unusual market rally, or that some risks were not sufficiently taken into account in calculating the standard deviation; I would advise you to approach such situations with great caution.
All this brings us to the last element of our conversation about the Sharpe ratio - namely, its limitations. They depend largely on the accuracy of the calculation of the standard deviation as a parameter representing the degree of risk exposure, as well as on the ability to use distributions of historical returns and volatility as a means of predicting future performance. As shown above, the limitations associated with calculating the standard deviation are due to the assumption that portfolio returns are normally distributed, which is not always the case. In addition, volatility patterns may not repeat - particularly in cases where volatility is calculated over shorter periods of time.
To illustrate the type of problems that can be associated with these restrictions, consider a portfolio in which nothing happens except the selling of deep-draw options that are very close to expiration. Because these options pay off in all but the most unlikely outcomes, portfolio managers using these strategies can achieve consistent returns with low volatility over long periods of time—often years. However, the corresponding Sharpe ratio masks the fact that from time to time, as a result of some sharp changes in the market, this portfolio will suffer significant losses. When this happens, we see both the limitations of calculating risk exposure and the risk associated with using historical returns as a means to predict future risks.
For these and other reasons, while the Sharpe ratio remains an important benchmark of risk-adjusted return, it is best used in conjunction with analytics that do not rely solely on standard deviation to measure risk, such as Period Return. Maximum Capital Drawdown (ROMAD).
Biography of William Sharp
Nobel Prize in Economics 1990 for his contributions to the theory of financial asset pricing
American economist William F. Sharp was born in Boston (Massachusetts). His parents were graduating from university at that time, his father with a degree in English literature, his mother with a degree in natural sciences. Then Sh.'s father worked at Harvard University. In 1940, in connection with his entry into the National Guard, the family moved to Texas and then to California. Sh. received his school education in Riverside (California). In 1951, he enrolled in medical school at the University of California at Berkeley, but a year later he became convinced that medicine was not his calling. He moved to the Los Angeles campus, choosing business management as his future major. During the first semester, Sh. studied accounting and economics - both courses were required to obtain a diploma in this specialty. Finding the course accounting boring, Sh. immediately became interested in microeconomics, which determined his future professional career. A particularly strong influence on him was exerted by university professors J.F. Weston, who taught finance and later attracted Sh. to work with G. Markowitz on a topic for which both would receive the Nobel Prize in the future, and A. Alchian, who taught economics. In 1955, Sh received a bachelor's degree in economics, and a year later, a master's degree.
After a short stay at military service Sh. began working as an economist at the RAND Corporation, where in those years developments were carried out in the field of game theory, computer technology, linear and dynamic programming and applied economics. Here Sh.’s joint work with G. Markowitz began on the problem of portfolio investments and the creation of a model reflecting the interrelationships of securities. While working in a corporation, Sh. defended his doctoral dissertation at the University of California at Los Angeles in 1961 on the “economics of transfer prices” (selling prices in payments between enterprises of the same company). In his dissertation, he explored a number of aspects of portfolio investment analysis based on the G. Markowitz model. Sh. called it a model with one coefficient; later it was called a one-factor model. The central idea of the dissertation was the position that income from securities correlates with each other only due to the influence of one common factor. The final chapter, “A Positive Theory of Security Market Behavior,” presented a one-factor model close to Sh.’s subsequent Capital Asset Pricing Model (CAPM).
In 1961, Sh. moved to teaching at the School of Business at the University of Washington in Seattle. For eight years he taught a wide range of subjects there, including microeconomics, financial theory, computer science, statistics, and operations research. In the process of teaching, Sh., in his own words, deepened his knowledge of the relevant sections economic theory. In 1963, in the journal Management Science, he first published a summary of the main ideas of his dissertation work in an article entitled “A Simplified Model for Portfolio Analysis.” At the same time, he continued to develop price model, which was outlined in the dissertation. As Sh. established, results similar to the analysis of a one-factor model can be obtained without taking into account the number of factors influencing income from securities. He discussed his new conclusion in January 1962 at the University of Chicago, and then presented in the article "Capital Asset Prices - A Theory of Market Equilibrium Under Conditions of Risk", published in 1964. It outlined the basis of the widely known stock price model capital, which was a step in the market analysis of the formation of prices for financial assets. Similar attempts further development G. Markowitz's models were undertaken in the mid-60s. J. Trainor, J. Lintner et al.
The model developed by Sh. was based on the assumption that an individual owner of shares (investor) may prefer to avoid risk through a combination of borrowed capital and an appropriately selected (optimal) portfolio of risky securities. In accordance with the Sh. model, the structure of the optimal portfolio of securities at risk depends on the investor’s assessment of future prospects various types securities, and not on his own attitude to risk. The latter is reflected only in the choice of a combination of risky stocks and investments in risk-safe securities (for example, Treasury bills), or in the preference for loans. For a shareholder who does not have special information relative to other shareholders, there is no reason to hold his share of the firm's equity in shares different from those held by other shareholders. Using the so-called “beta-value” indicator, the specific share of each shareholder in the total share capital of the company shows its marginal contribution to the risk of the entire market portfolio of risky securities. If the beta coefficient is greater than 1, then such shares have an above-average impact on the risk of the entire stock portfolio, and if the beta coefficient is less than 1, then the effect on the risk of the entire stock portfolio is below average. According to the Sh. pricing model, in efficiently operating capital markets, the risk premium and expected return from a security will change in direct proportion to the value of beta value. These relationships are associated with the formation of the equilibrium price in efficient capital markets.
Sh.'s model made it possible to determine, using the beta coefficient, the income expected from a security. It showed that risk could be transferred to the capital market, where it could be bought, sold and valued. Thus, the prices of risky securities are adjusted so that portfolio investment decisions become consistent.
The SH model is considered as a basis modern theory prices in financial markets. It has been widely used in empirical analysis, applied in practical research, and has become an important basis in decision-making practice in various fields economic life, primarily where the risk premium plays an important role. This applies to calculations of the cost of capital associated with making decisions on investment, mergers of companies, as well as in assessing the cost of capital as the basis for pricing in the field of regulated utilities, etc. Along with the portfolio investment model of G. Markowitz, the Sh. price model is included in all textbooks in financial economics.
In 1968, Sh. went to work on the campus of the University of California at Irvine to take part in the creation of the School of Social Sciences. For various reasons, this endeavor was not successful, and Sh. was invited to teach at the Graduate School of Business at Stanford University, where he moved in 1970. Shortly before that, he published the book “Portfolio Theory and Capital Markets” ", 1970), in which he outlined the main ideas of his theory of financial markets.
In the 70s Sh. focused his efforts on studying the problems associated with establishing equilibrium in capital markets, as well as its significance for the choice of an investment portfolio by the owner of shares. Then, from the mid-1970s, he turned to studying the role of investment policy for funds associated with pension provision. Written by him in the late 70s. the textbook "Investments" ("Investments", 1978; 2nd ed. 1985; 3rd ed. 1990) summarized a variety of empirical and theoretical material on this topic. A shortened version of the book entitled “Fundamentals of Investments” was published in 1989. While working on the textbook, Shch. supplemented his model by introducing into it a two-term procedure for choosing prices, which provided practical tools for assessing choice at there are several options. This model is widely used in practice.
Along with teaching and research work Sh. served as an investment consultant in a number of private firms, where he sought to put into practice some of the ideas of his theory of finance. He participated in assessing the reliability and risk of portfolio investments, selecting the optimal portfolio of securities, determining possible cash inflows, etc. Work at Merrill Lynch, Pierce and Smith and Wells Fargo enriched Sh. with real knowledge about investment practices.
In 1976-1977 Sh. was involved in work organized by the National Bureau economic research(NBER) group to study issues related to the adequacy of bank capital for the investment process. Sh. studied the relationship between deposit insurance and the risk of non-payment. The results of his work on the commission were summarized in five articles in the Journal of Financial and Quantitative Analysis in 1978.
At the end of the 70s. Sh. developed a fairly simple, but effective method finding solutions to a number of problems in portfolio investment analysis, which became widespread, despite the fact that the paper describing the solution mechanism - "An Algorithm for Portfolio Improvement" - remained unpublished until 1987.
In 1980, Sh. was elected president of the American Financial Association. In his acceptance speech, entitled "Decentralized Investment Management," he made several proposals to address the widespread practice among large investing institutions of dividing funds among professional investment managers.
In the 80s Sh. continued to deal with issues of investment planning policy for pension, insurance and other funds. He was particularly interested in the process of generating income in the market ordinary shares. The results of an empirical study of this issue were presented in the article "Some Factors Affecting the Income of Securities on the New York Stock Exchange, 1931-1979." ("Some Factors in New York Stock Exchange Security Returns, 1931-1979").
Sh. sought to implement the results of his research in training courses for training specialists in the placement of financial assets. In 1983, he helped Stanford University develop a one-week seminar on international investment management aimed at senior investment professionals. For three years, Sh. was one of the leaders of the program, and in subsequent years he continued to teach classes in this program. He helped create a similar three-week training program for a Japanese business school and taught there for five years.
In 1986, Sh. temporarily left Stanford University to organize his own research and consulting firm, Sharpe-Russell Research, whose purpose was to develop recommendations for insurance, pension, charitable and other funds and organizations for the placement of securities. It was supported by a number of American pension funds, The Frank Russell Company, as well as a group of professionals. In 1989, Sh. finally parted ways with teaching, resigning to devote all his time and energy to his company, which is now called William F. Sharp Associates. He remains a professor emeritus at Stanford University and continues to be involved in its academic life.
In the 70-80s. Sh. collaborated with many organizations and foundations involved investment activities. He is a trustee of the Research Foundation and the Education and Research Council of the Institute of Financial Analysts, a committee member of the Institute for Quantitative Research, and a consultant to the management department portfolio investments Swiss bank. For his services to research financial sector and his contribution to business education, Sh. was awarded by the American Assembly of Business Schools (1980) and the Federation of Financial Analysts (1989).
Sh. received the Alfred Nobel Prize in Economics for 1990 together with G. Markowitz and M. Miller “for their contribution to the theory of formation of the price of financial assets,” embodied in the so-called price model of equity capital.
Sh. is the father of two daughters, Deborah and Jonathan. In 1986 he remarried. His wife Katherine is a professional artist and is currently the administrator of Sh's family firm. In his spare time, Sh enjoys swimming, attending the opera, and football and basketball games.
Sharpe ratio calculation, calculation formula
Sharpe ratio shows the relationship between profitability and risk, namely, it measures the excess return of a portfolio per unit of risk. The higher the ratio, the higher the fund's historical return per unit of risk. The Sharpe ratio evaluates the degree to which equity approaches an exponential growth rate, or the degree to which income is stable. As the equity curve tends to exponential, Sharpe tends to infinity. In other words, Sharpe tends to infinity as all monthly returns tend to their own average value. It turns out that investors who focus on the Sharpe ratio strive to receive a stable income. Under the stability of income in in this case the constancy of profit is understood. If you try to get maximum income at a given risk, i.e. trade using optimal strategy profit, then the equity curve will not have an exponential shape with a constant growth rate. The average growth rate will change, this is the nature of the market - stable profitability is impossible in it. However, if you artificially limit the profitability, you can achieve a constant growth rate of the equity curve, and, accordingly, increase the Sharpe ratio. But such a measure will lead to a decrease in profits and the recovery factor. Trading in this case will not be optimal. Conclusion: Sharpe finds a stable trade, but, generally speaking, not an optimal one. Who should have more Sharpe? It is clear that the highest Sharpe value will be for those who strive to maximize this parameter. It is useless to compete with such people, and there is no need. I have never seen such people, and most likely they do not exist. Although someone suggested using the Sharpe ratio to evaluate traders in the Alpari competition. This is where we would see such people.
If the proposal had been accepted, the results of the competition would have been quite funny. However, it is interesting that if you do not use any machinations to increase this coefficient, then who will have more Sharpe? Obviously, intraday traders, especially pip traders, and portfolio traders. The smaller the trader’s timeframe, the more stable the profit over the months. Therefore, intraday traders have a chance to obtain a relatively large Sharpe value. Regarding portfolio trading, everything is also clear - diversification smoothes returns, bringing equity closer to the exponential. It will be most difficult for those who trade on one instrument and for the long term. Their Sharpe will be close to zero, unless the charts of the instruments being traded have a large Sharpe. The websites write that Sharp talks about the effectiveness of investments. And they even build fund ratings based on this ratio. In fact, it doesn’t say anything about efficiency. It only talks about the degree of profit stability. Stability is not efficiency; these concepts should not be confused. By comparing the Sharpe ratios of different funds, you can see which ones have more stable profits. If you do not pay attention to the profit itself, then you can consider a mutual fund with a yield of 12%, shown for the year from the moment of creation of this fund, to be the most effective investment money. Hence the conclusion: if you use the Sharpe ratio, it must be done in conjunction with such a parameter as annual profitability. By the way, banks have the largest Sharpe. If we consider the risk-free rate equal to zero, then their risk is in the thousands - an unattainable number for a trader. Banks resort to a method of artificially increasing this ratio - they redistribute profits. If the profit exceeds the fixed percentage, then they put the excess in their reserve.
If the profit does not reach the required percentage, they supplement it from the reserve, thereby ensuring stable payments. Like that. How to apply the Sharpe ratio A simple case: there are 2 funds, both have a return of 100% per year, while one of them has a low Sharpe, and the other has a high one. We can say with confidence that it is more profitable, and certainly psychologically calmer, to invest in a fund with the highest Sharpe value.
In this case, we will most likely make a profit in the first month, while in another fund, due to less stability, there may be no profit at first, or there may be a drawdown, however, as well as a very large profit in a short period of time. A fund with a higher Sharpe value will have profits more evenly distributed over the time period. It may be beneficial for a fund with a small Sharpe value to wait for some account drawdown or idle time. While there is no point in delaying investing in a fund with a high Sharpe - this will most likely contribute to the loss of possible profits. A high Sharpe of one of the funds may indicate better diversification, which indicates less risk. True, this same Sharp can talk about traders’ passion for pipsing. How this relates to risk is already difficult to assess.
Three problems with the Sharpe ratio
Although the Sharpe ratio is a useful measurement, it has a number of potential disadvantages.
1. Measuring profit in Sharpe ratio.
This measure—average monthly return (or return over another time period) expressed as an annual percentage—is more useful for assessing likely performance in the next month than for assessing performance throughout the year. For example, suppose that the manager makes 40% profit every month for six months, and the other 6 months bring him losses of 30%. Calculating the annual profit based on the monthly average, we get 60% (12 x 5%). However, if the position size is adjusted to match existing assets, as most managers do, the actual return for the year would be -11%. This will happen because for every dollar of assets held at the beginning of the period, there would only be $0.8858 left at the end of the period ((1.40)6 x (0.70)6 = 0.8858).
As this example shows, if you are concerned about evaluation potential profitability over an extended period, rather than just the next month or other interval, the measurement of profits used in the Sharpe ratio can lead to huge distortions. However, this problem can be circumvented by using the geometric mean (as opposed to the arithmetic mean) when calculating the average monthly return, which is then expressed as an annual percentage to give the numerator of the Sharpe ratio.
This assumes that trading assets are constant (profits are withdrawn and losses are made up). In other words, there is no reinvestment of profits and no reduction in investment in the event of losses. Generally speaking, while calculating returns including reinvestment is preferable, this is more than offset by the significant advantage of not having to evaluate minimum asset requirements in the event trading system. Moreover, a system with a higher profit, calculated without taking into account reinvestments, will most often demonstrate higher profits with them taken into account.
This section is adapted from J. Schwager, “Alternative to Sharpe Ratio Better Measure of Performance,” Futures, p. 57-58, March 1985.
The geometric average annualized return is exactly equivalent to the average annualized return including reinvestment, which is discussed later in this chapter in the section on the ratio of returns to maximum declines in asset values.
2. The Sharpe ratio does not differentiate between up and down movements in asset values. Sharpe ratio
measures volatility, not risk. And this is not necessarily the same thing.
In terms of the risk measure used in the Sharpe ratio, i.e. standard deviation of returns, up and down swings are considered equally bad. Thus, the Sharpe ratio would disfavor a manager who experiences sporadic, sharp increases in assets, even if the declines in asset values are small.
3. The Sharpe ratio does not differentiate between alternating and sequential losses. The measure of risk in the Sharpe ratio (standard deviation) does not depend on the sequence of winning and losing periods.
Sources for the article "Sharpe Ratio"
ru.wikipedia.org - free encyclopedia Wikipedia
aboutforex.biz - Forex website: simple about complex things
investpark.ru - investor portal
pifcapital.ru - website of mutual fund capital
dic.academic.ru - financial dictionary Academician
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In 1963, the American economist W. Sharp ( William Sharpe) proposed a new method for constructing the frontier of efficient portfolios, which allows one to significantly reduce the amount of necessary calculations. This method was later modified and is currently known as Sharpe single-index model.
The Sharpe model is based on the method of linear regression analysis, which allows you to relate two random variables - independent X and dependent Y - with a linear expression like Y = α + β X. In the Sharpe model, the value of some market index is considered independent. These could be, for example, the growth rate of gross domestic product, inflation rate, price index consumer goods and so on. Sharpe considered profitability as an independent variable r m, calculated based on the index Standard and Poor's(S&P500). Profitability is taken as the dependent variable r i some kind i th security. Since the S&P500 index is often considered as an index characterizing the securities market as a whole, the Sharpe model is usually called market model (Market Model), and the profitability r m– profitability of the market portfolio.
The Sharpe model examines the relationship between the return of each security and the return of the market as a whole. Basic assumptions of the Sharpe model:
- as profitability security is accepted mathematical expectation of profitability;
- there is a certain risk-free rate of return R f , that is, the yield of a certain security, the risk of which Always minimal compared to other securities;
– relationship deviationsreturn of a security from the risk-free rate of return(hereinafter referred to as the deviation of security yield) with deviationsprofitability of the market as a whole from the risk-free rate of return(hereinafter referred to as market return deviation) is described linear regression function;
– security risk means degree of dependence changes in the yield of a security from changes in the yield of the market as a whole;
– it is believed that the data past periods used in calculating profitability and risk fully reflect future profitability values.
According to the Sharpe model, deviations in security returns are associated with deviations in market returns by a linear regression function of the form
Where ( r i – R f) – deviation of the security’s yield from the risk-free one; ( R m – R f) – deviation of the market return from the risk-free one; α, β – regression coefficients.
Based on this formula, using the projected profitability of the securities market as a whole, it is possible to calculate the profitability of any security that constitutes it:
Where α i, β i– regression coefficients characterizing this security.
Theoretically, if the securities market is in equilibrium, then the coefficient α i will be equal to zero. But since in practice the market is always unbalanced, then α i shows excess return of a given security (positive or negative), i.e. the extent to which a given security is overvalued or undervalued by investors.
Coefficient β called β -risk, since it characterizes the degree of dependence of deviations in the profitability of a security from deviations in the profitability of the market as a whole.
The main advantage of the Sharpe model is that the interdependence of profitability and risk is mathematically substantiated: the more β – risk, the higher the profitability of the security.In addition, the Sharpe model has a peculiarity: there is a danger that the estimated deviation of the security's return will not belong to the constructed regression line. This risk is called residual risk . Residual risk characterizes the degree of dispersion of the deviation values of a security's return relative to the regression line. Residual risk is defined as the standard deviation of the empirical points of a security's return from the regression line. Residual risk i th security denote σ ri. The risk indicator of investing in a given security is determined β -risk and residual risk.
In accordance with the Sharpe model, the return on a securities portfolio is the weighted average of the return indicators of the securities and its components, taking into account β -risk. The portfolio return is determined by the formula
, (29)
Where Rf– risk-free return; Rm– expected profitability of the market as a whole.
The risk of a securities portfolio is determined by estimating the standard deviation of the function Rf and is determined by the formula
, (30)
Where σ m is the standard deviation of the profitability of the market as a whole, that is, an indicator of the risk of the market as a whole; βi, σ ri – β -risk and residual risk i th security.
At practical application The Sharpe model uses the following assumptions to optimize the stock portfolio:
– the yield on government securities, for example domestic government loan bonds, is taken as the risk-free rate of return;
– as the return on the securities market as a whole in period t, expert assessments market returns from analytical companies, from funds mass media etc. In a developed stock market, it is customary to use any stock indices for these purposes.
The main drawback of the model is the need to predict stock market returns and the risk-free rate of return. The model does not take into account fluctuations in risk-free returns. In addition, if the relationship between the risk-free return and the stock market return changes significantly, the model becomes distorted. Thus, the Sharpe model is applicable when considering a significant number of securities that describe b O most of the relatively stable stock market.
Questions
Define a securities portfolio.2. Name the methods for forming a securities portfolio
3. What risks of investing in securities do you know?
4. How is the expected return of a portfolio calculated?
5. Explain the mechanism for constructing the multi-index “return-risk” model by G. Markowitz.
6. How is the set of efficient portfolios based on a single index model determined?
Conclusion
Formation and development financial market Russia is carried out in difficult conditions and faces many problems of an objective and subjective nature. The market, which began its functioning “from scratch”, in the absence of knowledge, practical skills, business traditions and customs, was forced to adapt to a developed mechanism of financial intermediation, operating both in national and international frameworks. The globalization of financial markets, which is one of the key trends of the late twentieth century, means the need for Russia to comply with internationally accepted standards and “rules of the game.”
Redistribution occurs in the stock market with the help of financial intermediaries Money from investors to issuers based on securities. Depending on the types of financial institutions that perform the functions of financial intermediaries, banking, non-banking and mixed models of the securities market are distinguished. A mixed model is being formed in Russia.
Regulation of the securities market pursues a number of goals, the main of which is to create normal conditions for the functioning of the market itself and all its bona fide participants. Basic principle functioning of the securities market is the trust of investors in the market, which is ensured by maintaining honesty and openness on the part of market participants, preventing infringement of the rights of investors and market participants. The state, issuers and professional market participants are interested in ensuring investor confidence in the securities market. Ensuring investor confidence in the market is carried out through legislative and ethical standards.
Studying the development of the financial market as a mechanism for attracting financial resources into the sphere of real production, especially into the most promising sectors of the economy in terms of efficiency and profitability, will help young specialists make qualified decisions when placing free assets.
The securities market lives with expectations of the development of macroeconomic trends, political events, expectations associated with the performance of individual companies, with the actions of specific characters in the political-economic system.
Studying the mechanisms of circulation of securities in all their diversity will make it possible not only to predict dynamic market processes, but also to assess the prospects and trends of individual industries and the economy as a whole.
We continue the topic of market analysis and portfolio management. This time we will consider the topic of the index model of the famous American economist William Sharp (for which, by the way, he received the Nobel Prize in Economics in 1990). Today, the largest investment houses and funds in the world, as well as international banks They use this particular model to calculate the risks of investing in certain assets. I would like to note right away that the theoretical part of this model is quite difficult to master, so if you have any questions, you can ask them under the article or in the “ask an analyst a question” section.
Its essence is to simplify existing methods for constructing portfolios as much as possible in order to reduce the labor intensity of the process (sometimes even a whole staff of professional managers and financial analysts was not enough to construct a portfolio of securities using linear methods). In particular, this model uses regression analysis of the market - that is, analysis of historical quote data. It is clear that manual regression analysis of each asset from a total sample, which can reach up to several thousand, will require a very significant time, even with a large staff of competent employees, so back in the 60s Sharpe proposed using an index method of regression analysis to facilitate this process. The formula for calculating the Sharpe ratio is quite simple:
S=(R a -R f)/s a , where
R a – return on the direct asset;
R f – profitability of a risk-free investment;
s a – standard deviation of the asset.
In particular, the concept of beta coefficient was introduced, which has already been discussed a lot in many articles. The formula for calculating beta is well known to everyone: b= Cov am /s 2 m, where Cov am is the covariance of the asset return with the market, and s 2 m is the dispersion of the market return. This indicator indicates the degree of risk of investing in one or another. There is no point in describing this concept here for a long time, since the purpose of this article is different, and you can read more about calculating the beta coefficient in other articles on my blog. The essence of the Sharpe model is to use an already calculated index as a benchmark, on the basis of which the risk would be calculated. The general dependence of a security on the index is written as a formula:
r ia =a am +b am r im +e am , where
a am – bias coefficient (alpha coefficient);
b am – slope coefficient (beta coefficient);
e am – random error;
r ia – return on the asset for period i;
r im – market return for the same period.
According to Sharpe's theory, the beta coefficient indicates the dependence of the asset on market dynamics, and in turn, the alpha coefficient is the return of the asset regardless of the market index conditions. In the case of beta, it is assumed that this coefficient is static from period to period, and therefore, to calculate it, it is sufficient to use the ordinary linear regression method. The alpha coefficient, in turn, indicates overvaluation (in the case of positive alpha) or, on the contrary, undervaluation of a particular asset relative to the market (in the case of negative alpha).
Now we will try to summarize the material directly according to William Sharp’s model. So, the purpose of this model is to simplify linear methods construction of investment portfolios and regression analysis through the use of indices (that is, the return of a benchmark - a stock index or an individually constructed market index). To do this, regression analysis is carried out - that is, historical data on quotes of a specific asset and market are analyzed. In this case, the task is to identify the dependence of changes in the price of an asset on the dynamics of the benchmark and, based on this, ultimately calculate the risk coefficient, which will become an indicator of the relevance of investing in the asset. That's all. In one of the subsequent articles it will be posted specific example calculating the Sharpe ratio and using it directly when constructing a portfolio.
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